Transmission Line Equations

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Transmission lines take on many forms in order to accommodate particular applications. All rely on the same basic components - two or more conductors separated by a dielectric (insulator). The physical configuration and properties of all the components determines the characteristic impedance, distortion, transmission speed, and loss.

The following formulas are presented in a compact text format that can be copied and pasted into a spreadsheet or other application.

 

 

 

For the following equations, e is the dielectric constant  (e = 1 for air)

 

Two Conductors in Parallel (Unbalanced)
Above Ground Plane

For D << d, h

Z0= (69/e½) log10{(4h/d)[1+(2h/D)2]-½}

RF Cafe: Parallel conductors above a ground plane

Single Conductor Above Ground Plane
 

 

For d << h

Z0= (138/e½) log10(4h/d)

RF Cafe: Single conductor above a ground plane

Two Conductors in Parallel (Balanced)
Above Ground Plane

For D << d, h1, h2

Z0= (276/e½) log10{(2D/d)[1+(D/2h)2]-½}

RF Cafe: Parallel conductors above a ground plane

Two Conductors in Parallel (Balanced)
Different Heights Above Ground Plane

For D << d, h1, h2

Z0= (276/e½)log10{(2D/d)[1+(D2/4h1h2)]-½}

RF Cafe: Parallel conductors different heights above a ground plane

Single Conductor Between
Parallel Ground Planes

For d/h << 0.75

Z0= (138/e½) log10(4h/pd)

RF Cafe: Single conductor between parallel ground planes

Two Conductors in Parallel (Balanced)
Between Parallel Ground Planes

For d << D, h

Z0= (276/e½) log10{[4h tanh(pD/2h)]/pd}

RF Cafe: Two conductors between parallel ground planes

Balanced Conductors Between
Parallel Ground Planes

For d << h

Z0= (276/e½) log10(2h/pd)

RF Cafe: Balanced conductors between parallel ground planes

Two Conductors in Parallel (Balanced)
of Unequal Diameters

 

Z0= (60/e½) cosh-1 (N)

N = ½[(4D2/d1d2) - (d1/d2) - (d2/d1)]

RF Cafe: Parallel conductors - unequal diameters

Balanced 4-Wire Array

For d << D1, D2

Z0= (138/e½) log10{(2D2/d)[1+(D2/D1)2]-½}

RF Cafe: Balanced 4-wire array

Two Conductors
in Open Air

Z0= 276 log10(2D/d)

RF Cafe: Twol conductors in open air

5-Wire Array

For d << D

Z0= (173/e½) log10(D/0.933d)

 

 

RF Cafe: 5-wire array

Single Conductor in
Square Conductive Enclosure

For d << D

Z0≈ [138 log10(r) +6.48-2.34A-0.48B-0.12C]e-½)

A = (1+0.405r-4)/(1-0.405r-4)

B = (1+0.163r-8)/(1-0.163r-8)

C = (1+0.067r-12)/(1-0.067r-12)

r = D/d

RF Cafe: Single conductor in square conducting enclosure

Air Coaxial Cable with
Dielectric Supporting Wedge

For d << D

Z0≈ [138 log101(D/d)]/[1+(e-1)(q/360)]½)

e = wedge dielectric constant

q = wedge angle in degrees

RF Cafe: Air coaxial cable with dielectric supporting wedge

Two Conductors Inside Shield

For d << D, h

Z0= (69/e½) log10[(D2/2dh)(1-(h/D)4]

 

RF Cafe: Twin conductors inside shield

Two Conductors in Parallel (Unbalanced)
Inside Rectangular Enclosure

For d << D, h, w

                           ∞
Z0= (276/e½) {log10[(4h tanh(pD/2h)/pd)- ∑ log10[(1+mm2)/(1-nm2)]}
                            m=1

mm=sinh(pD/2h)/cosh(mpw/2h)

nm=sinh(pD/2h)/sinh(mpw/2h)

RF Cafe: Balanced 2-conductor line inside rectangular enclosure

 

Equations appear in "Reference Data for Engineers," Sams Publishing 1993